Упр.2.10 ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) 5^(x+1)+5^x+5^(x-1)=31;
2) 3^(x+1)-2·3^(x-1)-4·3^(x-2)=17;
3) 2^(x+2)-2^(x+1)+2^(x-1)-2^(x-2)=9;
4) 2·3^(2x+1)+3^(2x-1)-5·3^(2x)=36;
5) 6^(x-2)-(1/6)^(3-x)+36^((x-1)/2)=246;
6) 5·2^(x-1)-6·2^(x-2)-7·2^(x-3)=8^(x^2-1).

1. $$5^{x+1}+5^x+5^{x-1}=31$$

   $$5^x\cdot 5+5^x+5^x\cdot 5^{-1}=31$$

   $$5^x\left(5+1+\frac{1}{5}\right)=31$$

   $$5^x\cdot \frac{31}{5}=31$$

   $$5^x=5$$

   $$5^x=5^1,\quad x=1$$

2. $$3^{x+1}-2\cdot 3^{x-1}-4\cdot 3^{x-2}=17$$

   $$3^x\cdot 3-2\cdot 3^x\cdot 3^{-1}-4\cdot 3^x\cdot 3^{-2}=17$$

   $$3^x\left(3-\frac{2}{3}-\frac{4}{9}\right)=17$$

   $$3^x\cdot \frac{17}{9}=17$$

   $$3^x=9$$

   $$3^x=3^2,\quad x=2$$

3. $$2^{x+2}-2^{x+1}+2^{x-1}-2^{x-2}=9$$

   $$2^x\cdot 2^2-2^x\cdot 2+2^x\cdot 2^{-1}-2^x\cdot 2^{-2}=9$$

   $$2^x\left(4-2+\frac{1}{2}-\frac{1}{4}\right)=9$$

   $$2^x\cdot \frac{9}{4}=9$$

   $$2^x=4$$

   $$2^x=2^2,\quad x=2$$

4. $$2\cdot 3^{2x+1}+3^{2x-1}-5\cdot 3^{2x}=36$$

   $$2\cdot 3^{2x-1}\cdot 3^2+3^{2x-1}-5\cdot 3^{2x-1}\cdot 3=36$$

   $$3^{2x-1}(18+1-15)=36$$

   $$3^{2x-1}\cdot 4=36$$

   $$3^{2x-1}=9$$

   $$3^{2x-1}=3^2,\quad 2x-1=2,\quad x=\frac{3}{2}$$

5. $$6^{x-2}-\left(\frac{1}{6}\right)^{3-x}+36^{\frac{x-1}{2}}=246$$

   $$6^{x-2}-6^{-(3-x)}+36^{\frac{x-1}{2}}=246$$

   $$6^x\cdot 6^{-2}-6^x\cdot 6^{-3}+6^x\cdot 6^{-1}=246$$

   $$6^x\left(\frac{1}{36}-\frac{1}{216}+\frac{1}{6}\right)=246$$

   $$6^x\cdot \frac{41}{216}=246$$

   $$6^x=1296$$

   $$6^x=6^4,\quad x=4$$

6. $$5\cdot 2^{x-1}-6\cdot 2^{x-2}-7\cdot 2^{x-3}=8^{x^2-1}$$

   $$5\cdot 2^x\cdot 2^{-1}-6\cdot 2^x\cdot 2^{-2}-7\cdot 2^x\cdot 2^{-3}=2^{3(x^2-1)}$$

   $$2^x\left(\frac{5}{2}-\frac{6}{4}-\frac{7}{8}\right)=2^{3x^2-3}$$

   $$2^x\cdot \frac{1}{8}=2^{3x^2-3}$$

   $$2^x=2^{3x^2-2}$$

   $$x=3x^2-2$$

   $$3x^2-x-2=0$$

   $$(3x+2)(x-1)=0$$

   $$x=-\frac{2}{3}\quad \text{или}\quad x=1$$

Ответ

1) $$1$$; 2) $$2$$; 3) $$2$$; 4) $$\frac{3}{2}$$; 5) $$4$$; 6) $$-\frac{2}{3},\,1$$.